Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,21)(16,18)(17,31)(19,23)(20,24)(22,33)(25,41)(26,29)(27,37)(28,38)(30,35)(32,44)(34,40)(36,42)(39,43);(2,4)(6,10)(11,12)(13,15,20,28)(14,21,24,38)(16,29,23,37)(17,32,35,42)(18,26,19,27)(22,39,40,41)(25,33,43,34)(30,36,31,44);(5,7,9,8)(6,11,10,12)(13,16,19,24,20,23,18,14)(15,21,26,37,28,38,27,29)(17,30,34,40,35,31,33,22)(25,36,32,39,43,44,42,41);(1,2,3,4)(13,17,18,33,20,35,19,34)(14,22,23,31,24,40,16,30)(15,25,27,42,28,43,26,32)(21,36,29,41,38,44,37,39);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,16,24,23)(15,26,28,27)(17,33,35,34)(21,29,38,37)(22,30,40,31)(25,32,43,42)(36,41,44,39)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=hi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521678);